ASM : DE Lectures Di Topic 6 DE BUSN7( Di Horngre Di Lectures Di Standar De Mather FD 20 rs/TANG/Desktop/6013%20tutorial/2019Assignment2.pdf . Always show your full working. Unjustified correct answers may not receive full marks. Problem 1 Let a, B, and y be positive constants. Suppose that X and Y are jointly continuous random variables such that X ~ Beta(a + B, y) and the conditional density of Y given X is r(a + B) frix (y [ x) = ( @) I(B)x 2 pts (a) Fix x E (0, 1). Verify that the conditional density function frix (y | x) stated above is in fact a valid density function in y. 2 pts (b) Calculate the joint probability density function of X and Y, being sure to clearly specify the valid domain of points (x, y). 3 pts (c) Calculate the marginal probability density function of Y. Name the marginal distribution of Y as a known distribution and specify any parameter values. 2 pts (d) Write down the conditional probability density function fxy (X | Y) of X given Y. 3 pts (e) Compute the expected value of (1 - X)Y. (It may help to note that the joint density function found in (b) is a valid joint density for any choice of positive constants a, B, y.) Simplify your answer so that it does not use the gamma function I(.). 2 pts (f) Hence calculate Cov(X, Y). 4 pts (g) Let U1, . .., Un be independent Uniform([0, 1]) random variables. Denote by U() = min{U1, . .., Un; and U(n) = max (U1, . . ., Un} the first and last order statistics. If the correlation coefficient between Un) and U(n) is p = ", then what is the value of n and why? MagicBookProblem 2 Let Z1, Zz, ,.. . Z. be independent and identically distributed standard normal random variables. (As always, "standard" means having mean zero and variance one.) 3 pts (a) What are the constants ci, C2,... . C. such that the random variable M - ciZit cazz+... + caz. is independent of Z, # M for each i = 1,2.. n? In your answer, demonstrate why there is only one choice of the c,'s that has this property. 3 pts (b) Show that the following n random variables are all mutually independent: Di - ZIZZ D2 - 21 + 22 - 2Z3 Da - Zit Zz + Z3- 324 Da- 1 = Zit Zit Z, + . + 2 2+ 2.1- (2- 1)2. D. = Zit Zz + 23 +hit Z. 1 pt (c) Calculate Var( D, ) for 1 S j n -1. 2pts (d) For 1 s k 0, we have Y = 0 if and only if V = 0. Why, then, is g (0) = E[X | V = 0] not equal to ? ( You may draw a sketch to illustrate if you wish.) 2 pts (f) Now we consider a slightly different transformation: let W = YX. Calculate the joint probability density function fx,w(x, w) of X and W, being careful to specify the correct region of input points (x, w). 2 pts (g) Calculate the conditional probability density function fxw (x | w) of X given W. What happens at w = 0? 2 pts (h) Calculate the conditional expectation function h(w) = E[X | W = w] = / x fxiw (x/w)dx. What should h(0) be defined as in order to make h continuous? 2 pts (i) Suppose that Z is a random variable that is always nonnegative (P{Z 2 0) = 1) but has expected value E[Z] = 0. Use the fact that E(Z) > CP(Z > c} for all c > 0 to show that P(Z = 0) = 1. MagicBook